You can either -
send of the retention now £105
Then recieve your certificate/retention,tax disc and logbook within two weeks.
When you pick up adam with your 14 plate on you will have to wait for the logbook to come back to you, then you can send the logbook off with the retention certificate, then two weeks later you will get a logbook back with your pvt plate on,and a new tax disc, you can then put your plate on, so will more than likely take 3-4 weeks from picking adam up, to having your pvt plate on.
The other option is if your part exchanging your old car for adam, tell vauxhall you want the plate transfered from yours to the new one £80, and they do it all! When my boyfriend did that they unscrewed his plates from his part exchange and screwed them to the new car there and then, as its a grey area (and that was porsche main dealer not a backstreet garage)
Hope that helps a little? Private plates are a pain really